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understanding of projectile motion that is greatly simplified by treating the vertical and horizontal motion separately, as if the other did not even exist.

Project 8
Target practice. Horizontal projectile—rolling off a table .
     
The Idea
     
    In this experiment, you will try to hit a target. But, to improve your odds, you can use the laws of physics to predict where a projectile will land. Your projectile will be a steel ball or a marble. The physical situation is very much simplified when the projectile is shot (or launched) in the horizontal direction only. In this project, we see how close you can get to the target using the laws of physics that describe how horizontal objects move under the force of gravity.
What You Need
     
steel ball or marble (to serve as a projectile)
inclined track to get the marble rolling (This can be a piece of grooved wooden molding or a ruler with a groove)
flat, smooth, horizontal table
stopwatch or other timer (wrist watch, cell phone)
cup (your target)
meterstick
optional: motion sensor (to measure velocity)
Method
     
Part 1: Find the velocity of the marble coming off the ramp
     
    You will need this information to make your predictions.
Set up the ramp in such a way that its position remains fixed.
Place a marble at the top (or another arbitrarily mark) of the ramp.
Release the marble from the mark and measure the time it takes to go to the bottom of the ramp.
Repeat a few times until you get a consistent reading. Then, take the average. (If the ramp is too short or the slope is too great, it is more difficult to measure the time to go down the ramp.)
Find the final velocity at the bottom of the ramp using the equation:

Part 2
     
1. As we found in Project 5 , the vertical motion is independent of the horizontal, so we can determine the time it takes the marble to fall from the table just from the height,
h
, of the table. This is given by the equation:

2. Now make your prediction for how far the marble will go using R = vt. The distance the ball will go is now given by R = vt. Use the
v
you figured in Step 5 above and
t
from the previous step.
3. Set the (center of the) cup at the distance you predicted and try it out. No cheating. It is more fun to call your shot first, and then see if it works. Line the cup up visually, so it ison a straight line with the motion of the marble, as shown in Figure 8-1 .
Expected Results
     
    Clearly the expected result is for you to have the marble roll into the cup. If the marble hits at about the distance of the cup, but to the left or right, that should count as a hit. Hitting the target requires accurate measurement of the marble’s velocity on the table. It is reasonable to assume that the marble does not have any significant velocity loss for the short time it is rolling on the table.
    (A simpler way of doing this—appropriate for younger readers—is to
qualitatively
compare the distance the marble goes with the height of the ramp and skipping the math. The higher the ramp, the faster the marble and the farther it goes.)
    The time it takes to fall from a given distance is provided by the equation:

    To use this equation, the distance the projectile falls must be compatible with the units for gravitational acceleration,
g
. If you use 9.8m/s 2 for
g
,
h
must be in meters. The time to fall a given distance is shown in the following Table 8-1 :
    Table 8-1

    Using this table, the distance the projectile goes is simply its velocity multiplied by the time it is in the air (from the table or equation).
Why It Works
     
    The horizontal velocity of the marble is constant and unaffected by the fact that the marble is falling. The distance it moves is simply the horizontal velocity multiplied by the time.
    The time it takes to fall a given distance is dependent
only
on the vertical
distance
.
    Find the velocity at the bottom of a ramp using the fact that the final velocity is twice the average velocity divided by the time.

    Figure 8-1
Horizontal